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Properties

Label 4.4.4913.1-16.1-b4
Base field 4.4.4913.1
Conductor norm \( 16 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 4 \)
Rank \( 0 \)

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Base field 4.4.4913.1

Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 6 x^{2} + x + 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 1, -6, -1, 1]))
 
gp: K = nfinit(Polrev([1, 1, -6, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, -6, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(-\frac{1}{2}a^{3}+a^{2}+3a-\frac{3}{2}\right){x}{y}+\left(\frac{1}{2}a^{3}-2a-\frac{1}{2}\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}+3a+\frac{3}{2}\right){x}^{2}+\left(-11a^{3}-15a^{2}+69a-26\right){x}-\frac{79}{2}a^{3}-94a^{2}+227a-\frac{167}{2}\)
sage: E = EllipticCurve([K([-3/2,3,1,-1/2]),K([3/2,3,0,-1/2]),K([-1/2,-2,0,1/2]),K([-26,69,-15,-11]),K([-167/2,227,-94,-79/2])])
 
gp: E = ellinit([Polrev([-3/2,3,1,-1/2]),Polrev([3/2,3,0,-1/2]),Polrev([-1/2,-2,0,1/2]),Polrev([-26,69,-15,-11]),Polrev([-167/2,227,-94,-79/2])], K);
 
magma: E := EllipticCurve([K![-3/2,3,1,-1/2],K![3/2,3,0,-1/2],K![-1/2,-2,0,1/2],K![-26,69,-15,-11],K![-167/2,227,-94,-79/2]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2)\) = \((1/2a^3-a^2-2a+5/2)\cdot(-a-1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 16 \) = \(4\cdot4\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-416a^3+448a^2+2432a-928)\) = \((1/2a^3-a^2-2a+5/2)^{12}\cdot(-a-1)^{6}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 68719476736 \) = \(4^{12}\cdot4^{6}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{203862548967}{4096} a^{3} + \frac{203862548967}{4096} a^{2} + \frac{1019312744835}{4096} a + \frac{318403919021}{4096} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{SU}(2)$

Mordell-Weil group

Rank: \(0\)
Torsion structure: \(\Z/2\Z\oplus\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(\frac{21}{8} a^{3} - \frac{5}{2} a^{2} - \frac{43}{4} a + \frac{35}{8} : \frac{7}{8} a^{3} - \frac{29}{8} a^{2} - \frac{77}{8} a + \frac{25}{4} : 1\right)$ $\left(-\frac{1}{2} a^{3} - \frac{1}{4} a^{2} + \frac{7}{2} a - \frac{11}{4} : -\frac{9}{8} a^{3} + \frac{19}{8} a^{2} + \frac{53}{8} a - \frac{23}{8} : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 26.035750901914682695413858613181622022 \)
Tamagawa product: \( 72 \)  =  \(( 2^{2} \cdot 3 )\cdot( 2 \cdot 3 )\)
Torsion order: \(4\)
Leading coefficient: \( 1.67151100189927 \)
Analytic order of Ш: \( 1 \) (rounded)

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\((1/2a^3-a^2-2a+5/2)\) \(4\) \(12\) \(I_{12}\) Split multiplicative \(-1\) \(1\) \(12\) \(12\)
\((-a-1)\) \(4\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(2\) 2Cs
\(3\) 3B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6 and 12.
Its isogeny class 16.1-b consists of curves linked by isogenies of degrees dividing 24.

Base change

This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
\(\Q(\sqrt{17}) \) 2.2.17.1-4.1-a7
\(\Q(\sqrt{17}) \) a curve with conductor norm 1156 (not in the database)